We define a new notion of total curvature, called net total curvature, for
finite graphs embedded in Rn, and investigate its properties. Two guiding
principles are given by Milnor's way of measuring the local crookedness of a
Jordan curve via a Crofton-type formula, and by considering the double cover of
a given graph as an Eulerian circuit. The strength of combining these ideas in
defining the curvature functional is (1) it allows us to interpret the
singular/non-eulidean behavior at the vertices of the graph as a superposition
of vertices of a 1-dimensional manifold, and thus (2) one can compute the total
curvature for a wide range of graphs by contrasting local and global properties
of the graph utilizing the integral geometric representation of the curvature.
A collection of results on upper/lower bounds of the total curvature on
isotopy/homeomorphism classes of embeddings is presented, which in turn
demonstrates the effectiveness of net total curvature as a new functional
measuring complexity of spatial graphs in differential-geometric terms.Comment: Most of the results contained in "Total curvature and isotopy of
graphs in R3."(arXiv:0806.0406) have been incorporated into the current
articl